(Solved): letter b) 41.L. (This exercise assumes familiarity with the notion of the determinant of a square ma ...

41.L. (This exercise assumes familiarity with the notion of the determinant of a square matrix.) Let $L?\mathcal{L}\left({\mathbf{R}}^p,{\mathbf{R}}^p\right)$ and let $\left[c_y\right]$ be the matrix representation of $L$ with respect to the standard basis in ${\mathbf{R}}^p$. It is shown in linear algebra that $L$ is invertible if and only if $?=\det \left[c_{11}\right]$ is not zero. Furthermore, if $??=0$, then the matrix of $L^{?1}\mathrm{h}$ the form $\left[p_{ij}/?\right]$, where the $p_{ij}$ are polynomials in the $c_{ij}$. (a) Show that if $L_0$ is invertible and if ${?L?L_0?}_{pp}$ is sufficiently small, then $L$ is invertible. (b) Show that if $L_0$ is invertible, then the map $L?L^{?1}$ is continuous on a neighborhood of $L_0$ with respect to the norm in $\mathcal{L}\left({\mathbf{R}}^p,{\mathbf{R}}^p\right)$.